The first blast took place on Saturday in the rear of a car parked near a cash machine of the Samba Financial Group, formerly known as Saudi American Bank.
An official from the bank said the explosion had "not affected" the bank's operations and gave no further details.
US embassy spokeswoman in the Saudi capital Riyadh, Carol Kalin, said she was investigating the report but was unable to confirm if there had been an explosion outside the bank.
Al-Arabiya TV said the driver was wounded and taken to hospital where he was reportedly under guard.
Several other fighters fled the scene in a stolen vehicle to a neighbourhood where the US consulate is located and took shelter in an uninhabited house about 700 metres from the consulate.
Sources told Aljazeera police had laid siege to the house and sealed off the area.
Later on Saturday, a small explosion took place in a car outside a Saudi British Bank branch in Jidda, security sources said.
There was no immediate word on casualties in the blast which was also near a shopping mall, they said.
Insurgents have been waging a campaign of deadly bombings and attacks to destabilise the pro-US monarchy.